J00768

ARMA (Autoregressive Moving Average) Model for

Prediction of Rainfall in Regency of Semarang - Central

Java - Republic of Indonesia

Adi Nugroho1_{, Bistok Hasiholan Simanjuntak}2

1

Faculty of Information Technology, Satya Wacana Christian University

Salatiga, Central Java, Republic of Indonesia

2

Faculty of Agriculture and Business, Satya Wacana Christian University

Salatiga, Central Java, Republic of Indonesia

Abstract

Water is the main factor in determining the success of the activities of food crops, horticulture, and plantation. The main source of the water for agriculture and plantation comes from rainfall. This condition also occurs in regency of Semarang, Central Java, Indonesia. Therefore rainfall prediction will play an important role in the success of the activities. Univariate time series model of ARMA (Autoregressive Moving Average) can be used to predict it in the future. Data used in the study are taken on a monthly basis during the period from 2001 to 2013. The results showed that the prediction is quite accurate using method of ARMA in the study area.

Keywords: Rainfall Prediction, ARMA, Univariate Time Series

1 Introduction

Indonesia is tropical country. It has high enough

rainfall because its islands are surrounded by a vast

ocean which has a fairly high daily temperature and

humidity [8][10]. Currently, there are about 40.6

million hectares of agricultural and plantations areas

in this country [10] which mostly rely on the

availability of water depend on rainfall. In this

regard, the western and northeastern part of Indonesia

derived from volcanic activity. Agricultural or

plantation can be done as long as there is enough

water coming from rainfall [7][9].

The regency of Semarang – Central Java (area of the

study) is located on the island of Java, western part of

Indonesia (Figure 1). It is 6 º, 5 ' - 7 º, 10' South

Latitude and 110 º, 34 ' - 110 º, 35' East Longitude

with a total area of 37,366,838 hectares,

approximately 373.7 km2 [7].

Agriculture and plantation are the main sector that

supports the economy of the regency. In general, the

rainfall follows two kinds of seasons, a dry (April to

September) and a rainy season (October to March)

[8]. The study try to predict the monthly rainfall in

the next one year (2014) based on data of the rainfall

taken along 13 years earlier (2001-2013). The

ARMA model is used to predict of seasonal and

repeated rainfall because the data is stationary.

2.

Techniques of Prediction Using

ARMA Method

Time series is basically a measurement data taken in

chronological order within a certain time [4]. The

ARMA (Autoregressive Moving Average) method is

used in this study is because the characteristic of each

cascading is stationary (has a mean and constant

variance also covariance lag that does not depend on

where the calculation is done) [2]. The method is

also called the Box-Jenkins method as developed by

George Box and Gwilym Jenkins in 1976 [4].

The ARMA model consists of two parts, an

autoregressive (AR) part and a moving average (MA)

part. AR model can be written as follows [5].

… ԑ

… (1)

While the MA models can be written as follows [5].

ԑ ԑ … ԑ ԑ

coefficients of MA model.

 , , , … , are constants and

coefficients of AR model

An ARMA model requires stationary value. The

stationary can be tested using the ADF test

(Augmented Dickey Fuller) with this pattern [1][3].

∑ ԑ … (3)

is used to determine whether or not the

roots of the unit (unit root) with the

following hypothesis.

H0 : θ = 0 (the data contain unit roots) (not

stationary).

H1 : θ < 0 (the data do not contain unit

roots) (stationary).

 p is the lag in the autoregressive process.

 ԑ is the magnitude of the error or often referred to as white noise which is assumed

yt-1 and constant variance of σ2 or equal to 0

[4].

Table 1. ACF and PACF Pattern [5]

ACF PACF ARIMA

(p, 0, q)

Tend to zero after lag q.

Tend to zero after lag p.

In practice, the ARMA is often treated as an ARIMA

(Autoregressive Integrated Moving Average) with no

need for differencing process because the data is

stationary. In other words , the ARMA model can be

written as ARIMA (p, d, q ) which is more common

where p is the order of the autoregressive process, q

is the order of the moving average process, and d is

the differentiation process in the case of ARMA is 0,

so ARMA models are often written as ARIMA (p , 0

, q).

In this case, the values of p and q can be

predicted using plots values of ACF

(Autocorrelation Factor) and PACF (Partial

Autocorrelation Factor) as shown in Table 1 . ACF

and this PACF is defined as follows [6].

∑ … (4)

Where yk is the observed value, y is the mean, k is

the number of parameters, and n is the number of

times of observation. Whereas, the PACF is defined

as the following equation [4].

, ) … (5)

A prediction should be tested and evaluated to assess

its feasibility. In this paper, to assess the feasibility of

a predictive model, the calculation used AIC

(Aikake's Information Criterion), which is defined

using the following equation [6].

... (7)

Where and SSE =

Where yk is the observed value, y is the mean, k is

the number of parameters, and n is the number of

times of observation. In this case, it can be stated that

the smaller the AIC value calculation, meaning a

model that is taken is the best model [6].

After we got optimal values of p and q, then do linear

regression (OLS-Ordinary Least Square). We can get

the values of a and b in equation (1) and (2). Next,

find a model that can represent the ARMA time

series observation, with the functions, we predict it.

However, it also should be tested its accuracy. The

best way to evaluate the accuracy of forecasting is to

draw a graphic the results of observation values with

the values of the results of forecasting or,

mathematically, the model can also be evaluated by

Figure 2. Plot of Rainfall in Semarang Regency

Figure 3. Plot of ACF

 Calculate MAE (Mean Absolute Error).

∑ | |

… (8)

 Calculate MAPE (Mean Absolute Percentage Error).

∑ | |∗ % … (9)

Where μ is mean in point-to-t and a good model will

have a value of MAE and MAPE as small as possible

(less than or equal to 10 %) [3] .

3. Research and Discussion

Plot of rainfall in figure 2 is the original data plot in

the Semarang regency which shows relatively

stationary data. It has same deviation along it. This is

supported by the calculation of ADF as -7.3585,

which is indicated rainfall time series in Semarang

regency doesn’t have unit root. It is concluded to be

stationary. In this case, because the data is stationary

so ARIMA model (p, 0 , q) can be used. Next step is

how to find the value of p and q by pay attention to

the ACF and PACF plot and consider to AIC value.

Figure 3. Plot of ACF

Figure 4. Plot of PACF

Table 2. AIC calculations for the combination of ARIMA (p, d, q)

Model AIC

ACF and PACF plots in Figure 3 and Figure 4, based

on Table 1, indicating the possibility that ARIMA

models (6, 0, 3) is the best model for the p-value can

be approximated by PACF plots intersecting

horizontal line on the 6th lag and the value of q ACF plot can be approximated by horizontal lines

intersecting at the 3rd lag. However, to be sure, we need to do calculations AIC for models nearby. The

AIC calculations are shown in Table 2 above, where

these calculations (values shaded) consistent with the

ACF and PACF plots that provide a signal that the

ARIMA model (6, 0, 3) is the best model.

Based on the ARIMA model (6, 0, 3) , the calculation

of linear regression (OLS-Ordinary Least Square),

ARMA function is obtained as follows.

. . .

. .

. .

. ԑ . ԑ

. ԑ

Table 3. Value Rainfall Prediction for 2014

Jan 416.6

Furthermore, using the ARMA function above, we

can perform forecasting precipitation values in 2014

were the results as shown in Table 3. For the record,

the predicted values have a value of MAE 19.45714

and MAPE 9.581951 % , so it can be said that the

ARIMA model (6, 0 ,3) has a fairly good accuracy

(MAE relatively small and MAPE less than 10%) .

Generally, forecasting rainfall in the study area are

also in accordance with the recognized pattern 2

seasons, namely summer (April to September) and a

rainy season (October to March).

4 Conclusion

Based on the research results several conclusions can

be drawn as follows.

 On the basis of the monthly precipitation patterns in 2001 - 2013, visual observation

of the ACF and PACF plots and calculation

of the AIC, the rainfall in the district of

Semarang has ARIMA models (6, 0, 3).

 Based on the Box - Jenkins method of ARMA, then using monthly rainfall data in

2001 - 2013 in Semarang Regency

prediction can be done monthly rainfall for

the area in question in 2014.

 Based on the form and function of the ARMA, MAE and MAPE values are good

enough then ARMA model has fairly good

accuracy for prediction of rainfall the

following year (2014).

 The results of forecasting using ARMA model will be very useful for agricultural

planning and/or estate in Semarang district

that outlines rely on the water needs water

from rainfall in the area concerned.

References

[1] Cowpertwait, Paul S.P., Andrew V. Metcalfe, Introductory Time Series with R, New York : Springer Science+Business Media , Inc. ., 2009.

[3] Joshua, “Analysis of Vector Autoregression (VAR) The Interrelationship Between GDP Growth and Employment Growth ( Case Study : Indonesia Year 1977 to 2006)”, MS Thesis, Department of Mathematics, University of Indonesia, Jakarta, Republic of Indonesia, 2007.

[4] Lutkepohl, Helmut, New Introduction to Multiple Time Series Analysis, Berlin:Springer Science + Business Media, Inc., 2005.

[5] Sadeq, Ahmad, “Stock Value Prediction with ARIMA Model”, MS Thesis, Department of Management, University of Diponegoro, Semarang, Indonesia, 2008. [6] Schumway, Robert H., David S. Stoffer, 2011, Time Series Analysis and Its Application, New York : Springer Science +Business Media, Inc., 2011.

[7] Geography, topography, and geology Semarang District. http://www.semarangkab.go.id/utama/selayang-pandang/kondisi-umum/geografi-topografi.html. Retrieved July 10, 2013.

[8] Dry season and the rainy season in Indonesia. http://www.bmkg.go.id. Retrieved July 11, 2013.

[9] Site Agricultural Research and Development-Department of Agriculture, Republic of Indonesia. http://bbsdlp.litbang.deptan.go.id/tamp_komoditas.php . Retrieved July 20, 2013.

[10] The area of farms and plantations in Indonesia. http://indonesia.go.id/en/potential/natural-resources. Retrieved July 25, 2013.

Adi Nugroho earned a Bachelor of Engineering (ST) of

Geological Engineering - Institute of Technology Bandung

(ITB) in Republic of Indonesia in 1993. He also earned a

Masters in Management Information Systems from the

Gunadarma University in Jakarta, Republic of Indonesia, in

2002. Currently he is trying to complete his doctoral studies

in Computer Science at the PhD Program, Gadjah Mada

University in Republic of Indonesia and a career as a

lecturer in the Faculty of Information Technology – Satya

Wacana Christian University, in Salatiga, Central Java,

Republic of Indonesia.

Bistok Hasiholan Simanjuntak earned a Bachelor of

Engineering (Ir) of the Faculty of Agriculture – Satya

Wacana Christian University in Salatiga, Republic of

Indonesia, in 1989. Master's degree (Master of Science)

degree in Soil Science obtained from Bogor Agricultural

University in Republic of Indonesia (1997) and his PhD in

Soil Science is obtained from Brawijaya University in

Malang, Republic of Indonesia (2007) . Currently he is a

faculty member in the College of Agriculture and Business –

Satya Wacana Christian University in Salatiga, Central

Java, Republic of Indonesia. Areas of expertise and areas of